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4.2 Collective states of many-anyon system The microscopic calculations of the degeneracy splitting for a pair of vortices are important for understanding the collective states of anyons arising on top of the non-Abelian parent state when many Majorana fermions (Ising anyons) are present [97, 98, 99, 92]. Essentially, the sign of the splitting favors certain fusion channel ( 1 or ψ in the terminology of Ref.[100]) when two vortices carrying Majorana fermions are brought together. These fusion channels correspond to having a fermion (ψ-channel when E + < 0) or no fermion (1-channel when E + > 0) left upon fusing of two anyons. For pedagogical reason we start with the dilute anyon density limit assuming that the average distance between Majorana fermions is large compared with the coherence length ξ. In this regime, the many anyon state of the system will resemble gas of weakly bound pairs of anyons formed out of two anyons separated by the smallest distance. Because of the exponential dependence of the energy splitting the residual “interactions” with other anyons are exponentially smaller and can be ignored. In this scenario the parent state remains unchanged. When the density of anyons is increased so that the average distance between them becomes of the order of the Majorana bound state decay length (coherence length ξ in p-wave superconductors or magnetic length l c in Quantum Hall states) the system can form a non-trivial collective liquid (Wigner crystal of anyons or some other incompressible liquid state). This question has been investigated in Refs. [101, 102, 103, 92]. Although our approach used to calculate energy splitting 72
eaks down in this regime and one should resort to numerical calculations for the magnitude of the splitting, we believe that qualitative form of the splitting will remain the same. It is interesting to discuss the collective state that forms in this regime. Remarkably, it was shown in Ref. [92] that depending on the fusion channel (i.e. sign of the splitting) the collective state of anyons may be Abelian or non- Abelian. This result was obtained assuming that the magnitude of the splitting is constant and the sign of the splitting is the same for all anyons (positive or negative). However, because of the prefactor changing rapidly with the Fermi wave length we expect the magnitude of the splitting energy to be random realizing random bond Ising model discussed in Ref. [104, 92]. Finally, we mention that our calculations above and all existing studies of interacting many-anyon systems treat host vortices as classical objects with no internal dynamics. This is a well-defined mathematical framework, which corresponds to the BCS mean-field approximation. In real superconductors, however, there are certainly corrections to it. The order parameter field, ∆(r, t), which describes a certain vortex configuration has a non-trivial dynamics and fluctuates in both space and time. At low temperatures, when the system is fully gapped, these fluctuation effects are suppressed in the bulk, but they are always significant in the vicinity of the vortex core, where the order parameter vanishes. This dynamics gives rise to an effective motion of a vortex as well as to the dynamics of its shape and the radial profile. The relevant length-scales of these effects certainly exceed the Fermi wave-length, which is the smallest length-scale in the problem in most realistic systems. Even if the vortex is pinned, e.g. by disorder, its motion can be 73
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ABSTRACT Title of dissertation: MAJ
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MAJORANA QUBITS IN NON-ABELIAN TOPO
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Acknowledgments First and foremost,
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Table of Contents List of Figures L
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List of Figures 1.1 Braiding of Maj
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Chapter 1 Introduction The subject
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closed throughout the path. If one
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its original configuration, we requ
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mathematics. It is easy to check th
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Here τ x is the Pauli matrix actin
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y constructing the ground state pro
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here and leaves its proof to Append
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the zero mode (see Chapter 3 for de
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In the case of Ising anyons, 2N vor
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find a realistic material in nature
Page 31 and 32: states near the Fermi circle. We fi
Page 33 and 34: pairing: ( ) p H = ψ † 2 2m −
Page 35 and 36: dimensional non-chiral Majorana mod
Page 37 and 38: is formed out of two fermions, does
Page 39 and 40: gapless and should not be neglected
Page 41 and 42: zero modes. We study a one-dimensio
Page 43 and 44: of possible paired states which is
Page 45 and 46: where d ≡ (d 1 , d 2 , d 3 ) = (R
Page 47 and 48: If the order parameter is proportio
Page 49 and 50: 0.015 T/t 0.01 Normal p x + ip y 0.
Page 51 and 52: trivial p x - and p y -states lead
Page 53 and 54: with the gap operator being ˆ∆ =
Page 55 and 56: 0.2 T/t 0.1 Normal f p x + ip y µ
Page 57 and 58: 2.3 Discussion and Conclusions In c
Page 59 and 60: Chapter 3 Majorana Bound States in
Page 61 and 62: satisfied for m = 0. The singlevalu
Page 63 and 64: We summarize this section by provid
Page 65 and 66: Here γ a and Γ a are 4 × 4 Dirac
Page 67 and 68: with λ = µξ/v. It has the follow
Page 69 and 70: where n = (R∆, −I∆) field des
Page 71 and 72: Chapter 4 Topological Degeneracy Sp
Page 73 and 74: ound states are usually splitted an
Page 75 and 76: sponding quasiparticle operator can
Page 77 and 78: Of particular importance is the sig
Page 79 and 80: [61]. Although analytical expressio
Page 81: eyond perturbation theory and the r
Page 85 and 86: 4.3 Conclusions In this chapter, we
Page 87 and 88: the fermion bath, and consequently
Page 89 and 90: tion for the reduced density matrix
Page 91 and 92: like phonons. However, such process
Page 93 and 94: We also notice that in this formula
Page 95 and 96: etween two topological p x + ip y s
Page 97 and 98: neglect the AC phase. The Hamiltoni
Page 99 and 100: which in principle lead to a strong
Page 101 and 102: We conclude that the geometric phas
Page 103 and 104: The first-order term vanishes due t
Page 105 and 106: apply to any realistic system where
Page 107 and 108: ound states are typically close to
Page 109 and 110: Chapter 6 Non-adiabaticity in the B
Page 111 and 112: minor modifications in the details,
Page 113 and 114: {α(T )} to {α(0)}: |α(T )〉 =
Page 115 and 116: educed to 2 N−1 by fermion parity
Page 117 and 118: Berry connection of this state then
Page 119 and 120: wavefunction is u n (r), v n (r). W
Page 121 and 122: asis. The most general form of BdG
Page 123 and 124: 6.2.1 Effect of Tunneling Splitting
Page 125 and 126: Because E + ∼ ∆ 0 e −R/ξ , |
Page 127 and 128: tunneling amplitudes between them a
Page 129 and 130: ˆΓ i = iˆγ i ˆξiˆη i , i =
Page 131 and 132: considerations, we should have β 1
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states: ν(ε) = 2ν 0 ε √ ε2
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necessary as a matter of principle
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Chapter 7 Majorana Zero Modes Beyon
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We then demonstrate that in a finit
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The standard Abelian bosonization r
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and define the chiral fields by ˜
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of (7.2) has a product of the Klein
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7.3 Stability of the Degeneracy We
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quence. The splitting is then propo
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where α = 1 2 (K + K−1 − 2). A
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effect of quasiparticle tunneling o
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Appendix A Derivation of the Pfaffi
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which allows further simplification
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List of Publications Publications t
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Bibliography [1] K. von Klitzing, G
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[46] S. Das Sarma, C. Nayak, and S.
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[92] A. W. W. Ludwig, D. Poilblanc,
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[133] M. V. Berry, Proc. R. Soc. Lo
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